All Questions
Tagged with limits-and-colimits universal-property
9 questions
3
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Universal property of 2-presheaves and pseudo/lax/colax natural transformations
For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr ...
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5
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Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
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9
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Adjoining a morphism to a finitely complete category
Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a ...
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Is totality a (large) cocompleteness condition?
A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category ...
- 10.6k
16
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Original reference for categories of presheaves as free cocompletions of small categories
It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A ...
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8
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Original reference for the Fam construction
For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...
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11
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How to understand adjoint functors?
I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good ...
- 1,093
9
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Is there a useful limit or co-limit of a diagram that has only a single object?
I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same:
terminal/initial objects,
...
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12
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1
answer
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Adjoint Functors as Initial Objects of Some Category
Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
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